Using reflectometry to minimize the dependence of fluorescence intensity on optical absorption and scattering

The total diffuse reflectance R T and the effective attenuation coefficient µ eff of an optically diffuse medium map uniquely onto its absorption and reduced scattering coefficients. Using this premise, we developed a methodology where R T and the slope of the logarithmic spatially resolved reflectance, a quantity related to µ eff , are the inputs of a look-up table to correct the dependence of fluorescent signals on the media’s optical properties. This methodology does not require an estimation of the medium’s optical property, avoiding elaborate simulations and their errors to offer accurate and fast corrections. The experimental demonstration of our method yielded a mean relative error in fluorophore concentrations of less than 4% over a wide range of optical property variations. We discuss how the method developed can be employed to improve image fidelity and fluorochrome quantification in fluorescence molecular imaging clinical applications.


USING REFLECTOMETRY TO MINIMIZE THE DEPENDENCE OF FLUORESCENCE INTENSITY ON OPTICAL ABSORPTION AND SCATTERING
The Supplemental Material is organized as follows: In Sec. 1, we show the relationship between the effective attenuation coefficient (  ) and the slope of the logarithmic values of the spatially resolved reflectance.In Sec.II, we describe the preparation, characteristics and optical property calculation of the phantoms used for generating and testing of the look-up table (LUT) used to correct fluorescence intensities (FIs).

Relationship between 𝝁𝝁 𝒆𝒆𝒆𝒆𝒆𝒆 and the slope of the logarithmic spatially resolved reflectance
We determine and compare the relationship between   and the slope of the logarithmic values of the spatially resolved reflectance (SRR or (), where  is the radial distance from the beam position on the sample surface), as adapted in our work (i.e.,  10 [()]), and following the diffusion approximation-based formulation of Patterson et al. [1] (i.e.,  10 [ 2 ()]).First, we calculated the SRR from a pencil beam impinging on a semi-infinite medium (refractive index,   =1.4) -with   and   ′ coefficients -from air.The SRR is modelled by the diffusion theory as in [2]: where: and   ′ =   +   ′ .Then, the slope of  10 [ 2 ()] and  10 [()] were calculated for  between 3 and 4 mm.Those radii are within the diffusion approximation (i.e.,  ≫ 1   ′ ⁄ ). Figure S1 shows the results of the calculations and analysis.The comparison of  10 [ 2 ()] and  10 [()] for high and low scattering mediums is shown in Figure S1a.Unlike the slope of  10 [()], the slope of  10 [ 2 ()] is not always negative.The radial position of the stationary point of  10 [ 2 ()] depends on   ′ .Despite such differences, the slope of  10 [ 2 ()] and  10 [()] equally accounts for   , as shown in Figure S1b.Both slopes decrease as   increases.

Training phantoms
To prepare the training phantoms, Indian ink was initially dissolved in distilled water to prepare a 'dark' buffer solution.This solution was serially diluted to prepare six solutions with logarithmically distributed optical densities () at 670 nm as follows: 0.072, 0.144, 0.286, 0.572, 1.146 and 2.300.The ODs were measured using a spectrometer (USB4000 FL; Ocean Optics Inc., USA) in a transmission setup.  was calculated as a function of  by applying Beer-Lambert's law [3]: where [ ] is the natural logarithm operation and ℓ is the cuvette thickness for the measurements of absorbance.In our experience, ℓ ≡ 1.03 cm.On the other hand, a 'scattering' buffer solution of Intralipid (IL) at 16% was prepared and serially diluted to obtain six solutions with IL percentages (%) of: 16, 8, 4, 2, 1 and 0.5.  ′ of these solutions were calculated by using Staveren's formula [4], assuming a linear relationship between this coefficient and %: where  is the wavelength in microns (i.e.,  ≡ 0.670 µm).
Finally, 36 training phantoms were prepared by mixing equal parts -by weight -of the dark and scattering solutions.Alexa Fluor 680 was added according to the weight of the mixture.The characteristics of the training phantoms are listed in Table S1.  and   ′ of the phantoms were estimated to half of those coefficients calculated in the stock solutions, when ignoring the scattering of the Indian ink, the absorption of the IL, and the optical properties of the Alexa Fluor 680.The phantoms' preparation followed the procedure described above for the training phantoms.
In the first experiment, the fluorescence intensity correction was tested in phantoms with the same geometry but different concentrations of Alexa Fluor 680 ().Six and seven aqueous stock solutions of Indian ink and IL, respectively, were prepared with an approximated logarithmic distribution of  and %.Those solutions were mixed in equal parts by weight.Then, the Alexa Fluor 680 was added according to the weight of the mixture.The characteristics of the phantoms are listed in Table S2.  and   ′ were estimated as described before.
Table S2.Characteristics of the phantoms with the same geometry but different fluorophore concentration.In the second experiment, the fluorescence intensity correction was tested in phantoms with the same concentrations of Alexa Fluor 680 but different depths.Two aqueous stock solutions of Indian Ink and IL were prepared, by mixing them in equal parts -by weight -and adding the Alexa Fluor 680 according to the weight of the mixture.The characteristics of the phantoms are listed in Table S3.  and   ′ were estimated as described before.

Gel phantom
The gel phantom, composed of three regions, was prepared from three solutions by volume (one per phantom region).The compositions of the solutions are detailed in Table S4.The preparation of each solution was started by dissolving 450 mg of agar (Noble Agar; US Biological, United States) in water at 90°C (3.2°C above the melting point) using a magnetic stirring bar.IL and Indian ink were added while the temperature was reduced to 40°C (5.6°C above the gel point).Then, the Alexa Fluor 680 was added according to the weight of the solution.The mix was stirred while the temperature decreased, then transferred to the mold.  was roughly estimated as 0.10 cm -1 for solution 1, and 0.41 cm -1 for solutions 2 and 3. On the other hand,   ′ cannot be estimated by Equation S.3 due to the dependence of the scattering properties of IL on the agar content [5] and the mixing temperature and time [6].